Productivity

In addition to conducting our analysis with biomass as the measurement of ecosystem function, in this document we report our results using net primary productivity (NPP).

Mirroring the manuscript’s central analysis, our models for the across-treatment effect were encoded as: Productivity ~ -1 + Stage + Stage:Shannon, and the within-treatment effect was encoded as: Productivity ~ -1 + Richness:Stage + Richness:Stage:Shannon. All models successfully converged, with Rhat values of 1.0, and posterior predictive checks (PPC) were used to visually validate the model fits.

1 Figure 2 - Counter-gradient

The relationship between Shannon diversity and productivity was qualitatively similar to that of Shannon diversity for the majority of the models (4/6).

In Forest2, while the within-treatment slopes are consistent between biomass and productivity, the across-treatment slopes differ. During the with stage, the across-treatment slope is insignificant with biomass, but significant and slightly positive for productivity. During the without seed rain phase, the significant and negative across-treatment slope of the relationship between biomass and Shannon diversity becomes insignificant for productivity.

Dryland displays the most variation between the two measures of ecosystem functioning. The predominant difference is shows in the within-treatment slopes, as they flip from being significantly positive to significantly negative in both the with and without seed rain phases. Secondly, while the across-treatment slope is insignificant during the without seed rain phase for biomass as the ecosystem functioning, for productivity the slope is significant and positive. The reason for this change is clerical, because while seed biomass is incorporated into the productivity calculations, it is left absent from the total biomass calculations.

2 Figure 3 - Across-treatment effect

Considering the relationship between our measure of the internal coexistence processes within each model and the across-treatment effect of realized diversity on the focal ecosystem function (either productivity or biomass), we find that the aggregate patterns are nearly identical between ecosystem functions.

3 Figure 4 - Within-treatment effect

Considering the relationship between our measure of the internal coexistence processes within each model and the within-treatment effect of realized diversity on the focal ecosystem function (either productivity or biomass), we find that the aggregate patterns are nearly identical between ecosystem functions.

4 Model validation

This section of the document describes the statistical models’ validation, using Shannon diversity as the focal biodiversity metric and productivity as the focal ecosystem function.

Important terms:

4.1 Grass1

Clark, A. T., C. Lehman, and D. Tilman. 2018. Identifying mechanisms that structure ecological communities by snapping model parameters to empirically observed trade-offs. Ecology Letters 21:494–505.

4.1.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain                2.41      3.65    -4.92     9.36 1.00     2004     1992
## StageWithoutseedrain           -12.32      3.97   -20.18    -4.43 1.00     2208     1822
## StageWithseedrain:Shannon       17.73      1.52    14.85    20.79 1.00     2006     2205
## StageWithoutseedrain:Shannon    25.76      1.93    22.03    29.64 1.00     2217     2080
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    23.19      0.59    22.09    24.40 1.00     2991     2223
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.2964833 0.02290795 0.2506286 0.3416683

4.1.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

4.1.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.1.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

4.1.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.1.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                         Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain                82.98     26.46    32.04   133.75 1.00     5142     2639
## Ninitial4:StageWithseedrain               151.80     23.75   105.63   197.65 1.00     5072     2558
## Ninitial8:StageWithseedrain               166.37     20.19   125.25   205.31 1.00     5075     2456
## Ninitial16:StageWithseedrain              250.82     17.22   217.24   284.75 1.00     4946     2680
## Ninitial32:StageWithseedrain              278.86     19.96   240.30   319.25 1.00     5166     2906
## Ninitial2:StageWithoutseedrain             23.07     10.45     2.87    43.20 1.00     5316     2681
## Ninitial4:StageWithoutseedrain             64.50     14.48    36.02    93.12 1.00     5568     2782
## Ninitial8:StageWithoutseedrain             89.16     14.45    60.45   116.74 1.00     6188     2872
## Ninitial16:StageWithoutseedrain           184.60     18.03   148.68   219.84 1.00     5397     2686
## Ninitial32:StageWithoutseedrain           188.80     23.85   142.65   235.24 1.00     5119     2636
## Ninitial2:StageWithseedrain:Shannon       -36.63     16.47   -68.11    -4.97 1.00     5162     2743
## Ninitial4:StageWithseedrain:Shannon       -54.41     10.90   -75.70   -33.16 1.00     5090     2557
## Ninitial8:StageWithseedrain:Shannon       -47.37      7.60   -62.11   -32.12 1.00     5168     2396
## Ninitial16:StageWithseedrain:Shannon      -65.00      5.84   -76.68   -53.50 1.00     4913     2813
## Ninitial32:StageWithseedrain:Shannon      -64.85      6.32   -77.59   -52.75 1.00     5045     2980
## Ninitial2:StageWithoutseedrain:Shannon     -3.40      7.02   -17.33    10.27 1.00     5415     2744
## Ninitial4:StageWithoutseedrain:Shannon    -19.28      7.44   -33.91    -4.67 1.00     5533     2968
## Ninitial8:StageWithoutseedrain:Shannon    -23.94      6.41   -36.08   -11.31 1.00     6048     2895
## Ninitial16:StageWithoutseedrain:Shannon   -52.70      7.42   -67.30   -37.91 1.00     5485     2624
## Ninitial32:StageWithoutseedrain:Shannon   -44.05      9.10   -61.90   -26.51 1.00     5236     2616
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    15.05      0.43    14.22    15.91 1.00     6987     2815
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.6995809 0.01090739 0.6758253 0.7192806

4.1.2.1 Posterior predictive checks

4.1.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.1.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

4.1.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.2 Grass2

Turnbull, L. A., J. M. Levine, M. Loreau, and A. Hector. 2013. Coexistence, niches and biodiversity effects on ecosystem functioning. Ecology Letters 16:116–127.

4.2.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain               44.48      1.56    41.44    47.61 1.00     2172     2350
## StageWithoutseedrain            37.51      1.75    34.12    40.93 1.00     1867     1747
## StageWithseedrain:Shannon        8.72      0.55     7.63     9.80 1.00     2231     2550
## StageWithoutseedrain:Shannon    15.09      0.77    13.52    16.58 1.00     1826     2017
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    12.22      0.31    11.64    12.86 1.00     2838     2542
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.4418439 0.01928773 0.4022911 0.4776298

4.2.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

4.2.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.2.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

4.2.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.2.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                         Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain                85.32     32.40    22.50   147.16 1.00     4683     2998
## Ninitial4:StageWithseedrain               136.68     34.69    69.60   206.52 1.00     4808     2845
## Ninitial8:StageWithseedrain               133.48     32.48    68.65   198.41 1.00     5798     2701
## Ninitial16:StageWithseedrain               92.80     52.95   -12.42   197.98 1.00     4724     2071
## Ninitial32:StageWithseedrain               45.14     61.91   -78.54   168.23 1.00     4742     2505
## Ninitial2:StageWithoutseedrain            -16.38     22.60   -59.62    27.96 1.00     5166     2750
## Ninitial4:StageWithoutseedrain             27.19     13.72    -0.00    53.15 1.00     5140     3046
## Ninitial8:StageWithoutseedrain             38.70     10.52    18.48    60.01 1.00     4959     2496
## Ninitial16:StageWithoutseedrain            47.64     11.54    24.61    69.88 1.00     5143     2510
## Ninitial32:StageWithoutseedrain            72.50     14.94    43.38   101.49 1.00     5581     2919
## Ninitial2:StageWithseedrain:Shannon       -16.34     19.63   -54.07    21.95 1.00     4677     2986
## Ninitial4:StageWithseedrain:Shannon       -30.08     15.04   -60.46    -1.01 1.00     4833     2802
## Ninitial8:StageWithseedrain:Shannon       -20.22     11.21   -42.57     2.11 1.00     5820     2790
## Ninitial16:StageWithseedrain:Shannon       -4.79     14.93   -34.52    24.99 1.00     4707     2001
## Ninitial32:StageWithseedrain:Shannon        7.60     14.75   -21.65    37.07 1.00     4745     2408
## Ninitial2:StageWithoutseedrain:Shannon     45.98     13.98    18.37    72.55 1.00     5202     2542
## Ninitial4:StageWithoutseedrain:Shannon     22.09      7.44     7.96    36.93 1.00     5147     3020
## Ninitial8:StageWithoutseedrain:Shannon     17.69      4.76     7.96    26.96 1.00     5024     2553
## Ninitial16:StageWithoutseedrain:Shannon    12.15      4.22     4.05    20.49 1.00     5188     2645
## Ninitial32:StageWithoutseedrain:Shannon     3.28      4.45    -5.50    11.98 1.00     5562     2804
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     9.04      0.25     8.56     9.55 1.00     7003     3231
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5    Q97.5
## R2 0.4944095 0.01970995 0.4547698 0.530752

4.2.2.1 Posterior predictive checks

4.2.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.2.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

4.2.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.3 Grass3

May, F., V. Grimm, and F. Jeltsch. 2009. Reversed effects of grazing on plant diversity: The role of below-ground competition and size symmetry. Oikos 118:1830–1843.

4.3.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain               46.11      1.42    43.42    48.97 1.00     2118     2043
## StageWithoutseedrain            46.09      1.43    43.24    48.81 1.00     1781     1908
## StageWithseedrain:Shannon        2.69      0.53     1.62     3.71 1.00     2093     2070
## StageWithoutseedrain:Shannon     2.84      0.58     1.73     4.00 1.00     1831     1757
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     9.96      0.25     9.49    10.46 1.00     3213     2205
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##      Estimate  Est.Error       Q2.5      Q97.5
## R2 0.06514213 0.01638124 0.03489941 0.09834952

4.3.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

4.3.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.3.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

4.3.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.3.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                         Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain                17.56     37.59   -56.23    90.86 1.00     3080     2756
## Ninitial4:StageWithseedrain                -7.56     25.50   -56.56    42.17 1.00     3610     2794
## Ninitial8:StageWithseedrain                 4.11     25.57   -46.14    54.75 1.00     3611     2851
## Ninitial16:StageWithseedrain               58.76     23.15    12.86   102.98 1.00     3659     2877
## Ninitial32:StageWithseedrain               47.83     30.40   -10.57   106.73 1.00     3799     2901
## Ninitial2:StageWithoutseedrain             64.81      6.41    52.61    77.39 1.00     3175     2584
## Ninitial4:StageWithoutseedrain             25.46     10.98     3.20    46.72 1.00     3587     2592
## Ninitial8:StageWithoutseedrain             37.27     11.66    14.85    60.21 1.00     3429     2875
## Ninitial16:StageWithoutseedrain            64.81     21.39    24.68   107.89 1.00     3321     2812
## Ninitial32:StageWithoutseedrain            78.13     32.48    10.28   140.64 1.00     3474     2677
## Ninitial2:StageWithseedrain:Shannon        18.79     22.51   -25.01    62.97 1.00     3070     2721
## Ninitial4:StageWithseedrain:Shannon        25.33     11.16     3.49    46.73 1.00     3626     2821
## Ninitial8:StageWithseedrain:Shannon        17.35      9.05    -0.62    35.14 1.00     3617     2896
## Ninitial16:StageWithseedrain:Shannon       -1.04      6.86   -14.27    12.53 1.00     3658     2855
## Ninitial32:StageWithseedrain:Shannon        2.62      7.99   -12.86    17.96 1.00     3797     2985
## Ninitial2:StageWithoutseedrain:Shannon     -9.93      3.97   -17.78    -2.21 1.00     3164     2639
## Ninitial4:StageWithoutseedrain:Shannon     11.62      5.12     1.81    21.97 1.00     3581     2706
## Ninitial8:StageWithoutseedrain:Shannon      6.01      4.44    -2.91    14.43 1.00     3451     2810
## Ninitial16:StageWithoutseedrain:Shannon    -3.39      6.73   -17.07     9.20 1.00     3293     2920
## Ninitial32:StageWithoutseedrain:Shannon    -5.54      9.23   -23.25    13.63 1.00     3467     2659
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     6.73      0.19     6.37     7.14 1.00     5704     2920
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.2343049 0.02458069 0.1866864 0.2831217

4.3.2.1 Posterior predictive checks

4.3.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.3.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

4.3.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.4 Forest1

Rüger, N., R. Condit, D. H. Dent, S. J. DeWalt, S. P. Hubbell, J. W. Lichstein, O. R. Lopez, C. Wirth, and C. E. Farrior. 2020. Demographic trade-offs predict tropical forest dynamics. Science 368:165–168.

4.4.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain               31.42      2.02    27.35    35.36 1.00     1831     2150
## StageWithoutseedrain            27.84      2.31    23.02    32.36 1.00     1658     1561
## StageWithseedrain:Shannon        3.71      1.21     1.35     6.10 1.00     1803     2134
## StageWithoutseedrain:Shannon     3.83      1.71     0.51     7.30 1.00     1625     1523
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    13.35      0.34    12.69    14.03 1.00     2756     1912
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##      Estimate  Est.Error       Q2.5      Q97.5
## R2 0.04788901 0.01431746 0.02216935 0.07802648

4.4.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

4.4.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.4.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

4.4.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.4.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                         Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain                43.41      7.59    28.51    58.15 1.00     4655     3109
## Ninitial4:StageWithseedrain                48.90      6.71    35.63    62.21 1.00     4653     2803
## Ninitial8:StageWithseedrain                47.19      7.43    32.42    61.98 1.00     5048     2504
## Ninitial16:StageWithseedrain               37.01      9.66    18.04    55.13 1.00     4752     2817
## Ninitial32:StageWithseedrain               17.96     13.71    -9.50    44.75 1.00     4552     3196
## Ninitial2:StageWithoutseedrain             44.95      9.69    26.33    63.88 1.00     5308     2896
## Ninitial4:StageWithoutseedrain             42.39      6.11    30.34    54.14 1.00     5263     3186
## Ninitial8:StageWithoutseedrain             52.12      6.49    39.10    64.50 1.00     5084     2966
## Ninitial16:StageWithoutseedrain            18.26      6.38     5.82    30.71 1.00     4856     3189
## Ninitial32:StageWithoutseedrain            10.41      5.95    -1.15    22.12 1.00     5449     2661
## Ninitial2:StageWithseedrain:Shannon        -7.98      6.32   -20.26     4.49 1.00     4679     2863
## Ninitial4:StageWithseedrain:Shannon        -8.49      5.01   -18.40     1.61 1.00     4664     3033
## Ninitial8:StageWithseedrain:Shannon        -4.54      4.76   -14.13     4.75 1.00     5035     2286
## Ninitial16:StageWithseedrain:Shannon        1.80      4.86    -7.40    11.30 1.00     4803     2797
## Ninitial32:StageWithseedrain:Shannon        8.89      5.54    -1.93    20.17 1.00     4511     3306
## Ninitial2:StageWithoutseedrain:Shannon    -13.47      9.02   -30.99     3.68 1.00     5385     3007
## Ninitial4:StageWithoutseedrain:Shannon     -7.43      5.00   -16.93     2.33 1.00     5173     2955
## Ninitial8:StageWithoutseedrain:Shannon    -12.27      4.87   -21.61    -2.60 1.00     5114     2940
## Ninitial16:StageWithoutseedrain:Shannon    11.01      4.05     3.22    18.92 1.00     4895     3028
## Ninitial32:StageWithoutseedrain:Shannon    12.62      3.32     6.15    19.07 1.00     5619     2642
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    11.83      0.33    11.22    12.51 1.00     9144     2956
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.1468118 0.02249241 0.1039205 0.1911467

4.4.2.1 Posterior predictive checks

4.4.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.4.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

4.4.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.5 Forest2

Maréchaux, I., and J. Chave. 2017. An individual-based forest model to jointly simulate carbon and tree diversity in Amazonia: description and applications. Ecological Monographs.

4.5.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain               91.64      0.47    90.71    92.56 1.00     1944     1826
## StageWithoutseedrain            71.71      0.56    70.60    72.85 1.00     1898     2008
## StageWithseedrain:Shannon        0.56      0.18     0.21     0.91 1.00     1899     1773
## StageWithoutseedrain:Shannon     0.18      0.31    -0.43     0.79 1.00     1958     2087
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     3.56      0.09     3.39     3.75 1.00     2755     2165
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate   Est.Error      Q2.5     Q97.5
## R2 0.8971855 0.002267047 0.8922541 0.9011551

4.5.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

4.5.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.5.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

4.5.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.5.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                         Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain                98.58      3.75    91.17   105.85 1.00     4014     3000
## Ninitial4:StageWithseedrain                99.64      4.32    91.13   108.00 1.00     3891     3161
## Ninitial8:StageWithseedrain                94.86      6.31    82.29   107.20 1.00     4063     3178
## Ninitial16:StageWithseedrain               96.25     10.48    75.76   116.92 1.00     3846     2966
## Ninitial32:StageWithseedrain               92.48     15.41    62.57   123.22 1.00     3970     2828
## Ninitial2:StageWithoutseedrain             67.66      2.05    63.58    71.63 1.00     3997     2927
## Ninitial4:StageWithoutseedrain             68.56      1.94    64.72    72.31 1.00     3412     2810
## Ninitial8:StageWithoutseedrain             74.03      2.16    69.79    78.27 1.00     3846     2955
## Ninitial16:StageWithoutseedrain            66.26      1.97    62.36    70.06 1.00     4136     2993
## Ninitial32:StageWithoutseedrain            71.14      2.75    65.76    76.41 1.00     4301     2691
## Ninitial2:StageWithseedrain:Shannon        -4.08      2.47    -8.85     0.72 1.00     4025     2926
## Ninitial4:StageWithseedrain:Shannon        -2.67      2.04    -6.63     1.35 1.00     3889     3068
## Ninitial8:StageWithseedrain:Shannon        -0.34      2.35    -4.95     4.34 1.00     4055     3234
## Ninitial16:StageWithseedrain:Shannon       -0.86      3.13    -7.05     5.25 1.00     3857     3035
## Ninitial32:StageWithseedrain:Shannon        0.18      3.92    -7.66     7.78 1.00     3962     2849
## Ninitial2:StageWithoutseedrain:Shannon      3.78      1.51     0.93     6.83 1.00     3936     3016
## Ninitial4:StageWithoutseedrain:Shannon      3.06      1.21     0.71     5.39 1.00     3487     2969
## Ninitial8:StageWithoutseedrain:Shannon     -0.81      1.08    -2.93     1.37 1.00     3838     2763
## Ninitial16:StageWithoutseedrain:Shannon     2.68      0.93     0.82     4.49 1.00     4025     2990
## Ninitial32:StageWithoutseedrain:Shannon    -0.06      1.11    -2.21     2.14 1.00     4290     2693
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     2.97      0.08     2.81     3.15 1.00     8566     2769
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##    Estimate   Est.Error     Q2.5     Q97.5
## R2 0.927432 0.001620812 0.923994 0.9302469

4.5.2.1 Posterior predictive checks

4.5.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.5.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

4.5.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.6 Dryland

Reineking, B., M. Veste, C. Wissel, and A. Huth. 2006. Environmental variability and allocation trade-offs maintain species diversity in a process-based model of succulent plant communities. Ecological Modelling.

4.6.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain               63.06      1.52    60.22    66.03 1.00     1820     1865
## StageWithoutseedrain            43.42      1.93    39.61    47.32 1.00     1708     1886
## StageWithseedrain:Shannon        2.02      0.64     0.77     3.24 1.00     1809     1908
## StageWithoutseedrain:Shannon     4.59      1.21     2.15     6.98 1.00     1680     1857
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    10.80      0.28    10.28    11.36 1.00     2844     2214
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.3980418 0.02115576 0.3547234 0.4374282

4.6.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

4.6.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.6.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

4.6.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.6.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                         Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain               101.16      6.53    88.43   114.02 1.00     3930     3068
## Ninitial4:StageWithseedrain                95.49      8.93    78.25   113.11 1.00     4222     2459
## Ninitial8:StageWithseedrain               105.29     11.22    82.97   126.94 1.00     4199     2928
## Ninitial16:StageWithseedrain              103.48     15.79    72.81   134.61 1.00     3865     3118
## Ninitial32:StageWithseedrain              131.59     20.80    91.09   171.76 1.00     3806     3389
## Ninitial2:StageWithoutseedrain             50.33      4.25    41.89    58.59 1.00     4700     2948
## Ninitial4:StageWithoutseedrain             66.10      4.38    57.57    74.71 1.00     4729     2883
## Ninitial8:StageWithoutseedrain             63.96      4.60    54.93    73.12 1.00     3837     3192
## Ninitial16:StageWithoutseedrain            79.92      6.48    67.36    92.40 1.00     4083     3022
## Ninitial32:StageWithoutseedrain            77.88      9.93    58.27    97.24 1.00     3971     2813
## Ninitial2:StageWithseedrain:Shannon       -24.61      4.49   -33.41   -15.65 1.00     3925     3144
## Ninitial4:StageWithseedrain:Shannon       -14.02      4.67   -23.13    -4.99 1.00     4243     2648
## Ninitial8:StageWithseedrain:Shannon       -15.26      4.62   -24.20    -6.12 1.00     4184     2808
## Ninitial16:StageWithseedrain:Shannon      -11.49      5.35   -21.99    -1.17 1.00     3852     3122
## Ninitial32:StageWithseedrain:Shannon      -17.73      5.99   -29.25    -6.14 1.00     3820     3344
## Ninitial2:StageWithoutseedrain:Shannon     -3.34      3.37    -9.87     3.44 1.00     4693     2966
## Ninitial4:StageWithoutseedrain:Shannon    -10.32      3.00   -16.14    -4.54 1.00     4750     2913
## Ninitial8:StageWithoutseedrain:Shannon     -7.10      2.73   -12.58    -1.80 1.00     3807     3160
## Ninitial16:StageWithoutseedrain:Shannon   -13.87      3.47   -20.63    -7.02 1.00     4069     2967
## Ninitial32:StageWithoutseedrain:Shannon   -11.23      4.94   -20.82    -1.43 1.00     3991     2899
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     7.97      0.23     7.54     8.44 1.00     8386     2429
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.5791532 0.01669405 0.5439361 0.6099267

4.6.2.1 Posterior predictive checks

4.6.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.6.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

4.6.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.